632 BELL SYSTEM TECHNICAL JOURNAL 



separate channels. Since disturbances will be produced in many of 

 the channels when the applied voltage goes beyond the overload point, 

 it will be useful to know the fraction of the time that this may be ex- 

 pected to occur; this fraction will be called the overload expectation 

 and denoted by e. It is important to notice that this quantity e is not 

 necessarily the fraction of the time during which the performance of 

 the amplifier will be unsatisfactory. This might perhaps be the case 

 for a device having an instantaneous cutoff characteristic, but for an 

 ordinary amplifier the time constants (among other things) affect the 

 results of overloading. The interpretation of the overload expectation 

 will be discussed further later; consideration must be given first to 

 how it is obtained. 



The n- Channel Voltage Distribution 



The load in each channel is applied at voice frequency to the input 

 side of a modulator, the voice frequency instantaneous voltage distribu- 

 tion being as shown by the curve w = 1 of Fig. 2. The overloading 

 of the amplifier is determined, however, by the distribution of the sum 

 of n such voltages after each has been shifted by the modulator to the 

 appropriate carrier frequency, one side-band being suppressed. It 

 may be shown that if the phases of the various components of the voice 

 frequency input were random, the distribution of instantaneous voltage 

 at side-band frequency would be identical with that measured at voice 

 frequency. It is known, however, that the phases at voice frequency 

 are not entirely random, and there may thus be differences between 

 the two distributions. The results of a number of tests bearing upon 

 this point indicate that any error resulting from the use of the dis- 

 tribution measured at voice frequency will be small for systems of 

 few channels, and will rapidly disappear as the number of channels 

 is increased. 



Theoretically, the resultant w-channel voltage distribution can be 

 derived from the single-channel distribution by straightforward 

 analytical methods; in the present case, however, expression of the 

 result in useful form is very difficult because of the form of the single- 

 channel curve. This difficulty might be resolved by using graphical 

 or numerical methods, as applied later to the volume distribution 

 curves; fortunately, the fact that the voice frequency voltage distribu- 

 tions may be used throughout permitted the resultant w-channel 

 distributions to be obtained much more easily. Since the addition of 

 voltages from the several carrier channels does not depend materially 

 upon the frequencies at which the channels appear in the system, the 

 addition of n channels at voice frequency will give the desired n- 



